Burning down the house

In the last problem, you showed that the number of trees consumed by the average fire is given by $f\times\rho_\textrm{tree}\times L^2\times\langle s\rangle.$

The conservation relation shows that this is equal to the average number of trees grown per time step, $r\times\rho_\textrm{empty}\times L^2.$

From the normalization of the lattice densities, we know that $\rho_\textrm{empty} = 1 - \rho_\textrm{tree} - \textrm{burning},$ and if we assume that $\rho_\textrm{burning} \ll \{\rho_\textrm{empty}, \rho_\textrm{tree}\},$ then $\rho_\textrm{empty} = 1 - \rho_\textrm{tree}.$

If we equate the two sides of the conservation equation, and use the relation between $\rho_\textrm{empty}$ and $\rho_\textrm{tree},$ we find $\begin{aligned} f\times\rho_\textrm{tree}\times L^2\times\langle s\rangle &= r\times\rho_\textrm{empty}\times L^2 \\ \langle s\rangle &= \frac{r}{f}\frac{\rho_\textrm{empty}}{\rho_\textrm{tree}} \\ &= \frac{r}{f}\frac{1-\rho_\textrm{tree}}{\rho_\textrm{tree}}. \end{aligned}$